An enumeration process for racks

Abstract

Given a presentation for a rack $\mathcal R$, we define a process which systematically enumerates the elements of $\mathcal R$. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for $n$-quandles introduced by Winker. We provide pseudocode that is similar to that given by Holt for the Todd-Coxeter process. We prove that the process terminates if and only if $\mathcal R$ is finite, in which case, the procedure outputs an operation table for the finite rack. We conclude with an application to knot theory.